In this paper, we consider a Bitcoin system that consists of a set of miners (denoted as \(\mathcal {M}\) ) competing for mining blocks and a set of users (denoted as \(\mathcal {N}\) ) bidding transaction packaging. That is, a multi-miner and multi-user game is employed to depict the real Bitcoin system. Following the assumption made in [1]}, [2]}, we suppose each miner has one rig, and all rigs are assumed to be identical which means they share the same operation cost and computing power for simplicityNotably, our scheme is also applicable to the case where each miner has multiple or heterogeneous rigs. One can easily extend our model by appropriately adjusting \(\alpha _i(t)\)  and \(C_m(x_i,t)\)  defined in the following context.. Primary parameters used in this section are listed in Table 1.
